Explain how the 1st half of the story is necessary to understand why Connie is vulnerable in the second half of the story. If this story is divided into two parts, identify the break point
I think that the first part of the story lets the reader inside the world of Connie's mind: not the way things are, but the way she sees them. Everything is about what she wants and what she believes. Her mother tries to get her to see the world for what it is, but she rejects it: it doesn't fit with her sense of her beauty and that of the world she has created in her mind.
To avoid her mother and the world as she sees it, Connie is sneaky and manipulative, putting her mother's fears to rest so Connie can hang with an older crowd, something her mother would not knowingly let her do. But Connie's mom believes what Connie wants her to see, not what is real.
There is an irony here. Connie sees the world as she wants it to be, and even though her mother fights the appearance of the world with the realities of the world, Connie's mom comes also to see the world the way Connie wants her to see it. She believes her daughter's lies and tricks, and is lulled into a false sense of security. Connie has a false sense of security as well, but it is founded on the fact that she sees nothing to be afraid of as she dreams her dreams and plays pretend, as if she exists in a different world.
And then the family sets off to go to an afternoon barbeque, leaving Connie home alone.
When Arnold Friend arrives, Connie is startled. At first she is unsure of what to do. She has no experience in knowing what the world is really like because she has resisted all along what her mother has tried to tell her. Even as she starts to recognize the warning signals, they confuse her because she has experience only with the world as she wishes it to be, not as it truly is.
Arnold eventually talks Connie into just giving up. He uses the threat against her family to make her more vulnerable, but her confusion makes the threat secondary. Like a snake charmer hypnotizing a cobra, Connie cannot resist his voice, cannot resist what she sees as the inevitable. The will to fight leaves her and she resigns herself to her fate.
Had Connie had a better handle on the true world, perhaps she might not have been so vulnerable when faced with the ugly truth about life and the dangers it holds: that world is one she had never seen and could not, at last, even imagine.
If I had to identify the point that breaks the story into two parts, I don't think I would go with the obvious arrival of Arnold at her house. I think instead I would find the break where Connie distances her own behavior from that of the Pettinger girl (the girl with a questionable reputation): Connie infers that SHE would never be like "that dope," and then admits to feeling sorry that she so cruelly tricks her mother with her lies. The narrator informs the reader that once again her mom continues to try to bring Connie out of her fantasy world, but Connie's smooth and calculating lies seal her fate. The real cruelty will be her mother having to live with what happens to Connie.
Connie's mother kept dragging her back to the daylight by finding things for her to do or saying suddenly, 'What's this about the Pettinger girl?"
And Connie would say nervously, "Oh, her. That dope." She always drew thick clear lines between herself and such girls, and her mother was simple and kind enough to believe it. Her mother was so simple, Connie thought, that it was maybe cruel to fool her so much.